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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    package org.apache.commons.math3.analysis.polynomials;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import org.apache.commons.math3.analysis.differentiation.DerivativeStructure;<a name="line.19"></a>
<FONT color="green">020</FONT>    import org.apache.commons.math3.analysis.differentiation.UnivariateDifferentiableFunction;<a name="line.20"></a>
<FONT color="green">021</FONT>    import org.apache.commons.math3.exception.DimensionMismatchException;<a name="line.21"></a>
<FONT color="green">022</FONT>    import org.apache.commons.math3.exception.NoDataException;<a name="line.22"></a>
<FONT color="green">023</FONT>    import org.apache.commons.math3.exception.util.LocalizedFormats;<a name="line.23"></a>
<FONT color="green">024</FONT>    <a name="line.24"></a>
<FONT color="green">025</FONT>    /**<a name="line.25"></a>
<FONT color="green">026</FONT>     * Implements the representation of a real polynomial function in<a name="line.26"></a>
<FONT color="green">027</FONT>     * Newton Form. For reference, see &lt;b&gt;Elementary Numerical Analysis&lt;/b&gt;,<a name="line.27"></a>
<FONT color="green">028</FONT>     * ISBN 0070124477, chapter 2.<a name="line.28"></a>
<FONT color="green">029</FONT>     * &lt;p&gt;<a name="line.29"></a>
<FONT color="green">030</FONT>     * The formula of polynomial in Newton form is<a name="line.30"></a>
<FONT color="green">031</FONT>     *     p(x) = a[0] + a[1](x-c[0]) + a[2](x-c[0])(x-c[1]) + ... +<a name="line.31"></a>
<FONT color="green">032</FONT>     *            a[n](x-c[0])(x-c[1])...(x-c[n-1])<a name="line.32"></a>
<FONT color="green">033</FONT>     * Note that the length of a[] is one more than the length of c[]&lt;/p&gt;<a name="line.33"></a>
<FONT color="green">034</FONT>     *<a name="line.34"></a>
<FONT color="green">035</FONT>     * @version $Id: PolynomialFunctionNewtonForm.java 1422195 2012-12-15 06:45:18Z psteitz $<a name="line.35"></a>
<FONT color="green">036</FONT>     * @since 1.2<a name="line.36"></a>
<FONT color="green">037</FONT>     */<a name="line.37"></a>
<FONT color="green">038</FONT>    public class PolynomialFunctionNewtonForm implements UnivariateDifferentiableFunction {<a name="line.38"></a>
<FONT color="green">039</FONT>    <a name="line.39"></a>
<FONT color="green">040</FONT>        /**<a name="line.40"></a>
<FONT color="green">041</FONT>         * The coefficients of the polynomial, ordered by degree -- i.e.<a name="line.41"></a>
<FONT color="green">042</FONT>         * coefficients[0] is the constant term and coefficients[n] is the<a name="line.42"></a>
<FONT color="green">043</FONT>         * coefficient of x^n where n is the degree of the polynomial.<a name="line.43"></a>
<FONT color="green">044</FONT>         */<a name="line.44"></a>
<FONT color="green">045</FONT>        private double coefficients[];<a name="line.45"></a>
<FONT color="green">046</FONT>    <a name="line.46"></a>
<FONT color="green">047</FONT>        /**<a name="line.47"></a>
<FONT color="green">048</FONT>         * Centers of the Newton polynomial.<a name="line.48"></a>
<FONT color="green">049</FONT>         */<a name="line.49"></a>
<FONT color="green">050</FONT>        private final double c[];<a name="line.50"></a>
<FONT color="green">051</FONT>    <a name="line.51"></a>
<FONT color="green">052</FONT>        /**<a name="line.52"></a>
<FONT color="green">053</FONT>         * When all c[i] = 0, a[] becomes normal polynomial coefficients,<a name="line.53"></a>
<FONT color="green">054</FONT>         * i.e. a[i] = coefficients[i].<a name="line.54"></a>
<FONT color="green">055</FONT>         */<a name="line.55"></a>
<FONT color="green">056</FONT>        private final double a[];<a name="line.56"></a>
<FONT color="green">057</FONT>    <a name="line.57"></a>
<FONT color="green">058</FONT>        /**<a name="line.58"></a>
<FONT color="green">059</FONT>         * Whether the polynomial coefficients are available.<a name="line.59"></a>
<FONT color="green">060</FONT>         */<a name="line.60"></a>
<FONT color="green">061</FONT>        private boolean coefficientsComputed;<a name="line.61"></a>
<FONT color="green">062</FONT>    <a name="line.62"></a>
<FONT color="green">063</FONT>        /**<a name="line.63"></a>
<FONT color="green">064</FONT>         * Construct a Newton polynomial with the given a[] and c[]. The order of<a name="line.64"></a>
<FONT color="green">065</FONT>         * centers are important in that if c[] shuffle, then values of a[] would<a name="line.65"></a>
<FONT color="green">066</FONT>         * completely change, not just a permutation of old a[].<a name="line.66"></a>
<FONT color="green">067</FONT>         * &lt;p&gt;<a name="line.67"></a>
<FONT color="green">068</FONT>         * The constructor makes copy of the input arrays and assigns them.&lt;/p&gt;<a name="line.68"></a>
<FONT color="green">069</FONT>         *<a name="line.69"></a>
<FONT color="green">070</FONT>         * @param a Coefficients in Newton form formula.<a name="line.70"></a>
<FONT color="green">071</FONT>         * @param c Centers.<a name="line.71"></a>
<FONT color="green">072</FONT>         * @throws org.apache.commons.math3.exception.NullArgumentException if<a name="line.72"></a>
<FONT color="green">073</FONT>         * any argument is {@code null}.<a name="line.73"></a>
<FONT color="green">074</FONT>         * @throws NoDataException if any array has zero length.<a name="line.74"></a>
<FONT color="green">075</FONT>         * @throws DimensionMismatchException if the size difference between<a name="line.75"></a>
<FONT color="green">076</FONT>         * {@code a} and {@code c} is not equal to 1.<a name="line.76"></a>
<FONT color="green">077</FONT>         */<a name="line.77"></a>
<FONT color="green">078</FONT>        public PolynomialFunctionNewtonForm(double a[], double c[]) {<a name="line.78"></a>
<FONT color="green">079</FONT>    <a name="line.79"></a>
<FONT color="green">080</FONT>            verifyInputArray(a, c);<a name="line.80"></a>
<FONT color="green">081</FONT>            this.a = new double[a.length];<a name="line.81"></a>
<FONT color="green">082</FONT>            this.c = new double[c.length];<a name="line.82"></a>
<FONT color="green">083</FONT>            System.arraycopy(a, 0, this.a, 0, a.length);<a name="line.83"></a>
<FONT color="green">084</FONT>            System.arraycopy(c, 0, this.c, 0, c.length);<a name="line.84"></a>
<FONT color="green">085</FONT>            coefficientsComputed = false;<a name="line.85"></a>
<FONT color="green">086</FONT>        }<a name="line.86"></a>
<FONT color="green">087</FONT>    <a name="line.87"></a>
<FONT color="green">088</FONT>        /**<a name="line.88"></a>
<FONT color="green">089</FONT>         * Calculate the function value at the given point.<a name="line.89"></a>
<FONT color="green">090</FONT>         *<a name="line.90"></a>
<FONT color="green">091</FONT>         * @param z Point at which the function value is to be computed.<a name="line.91"></a>
<FONT color="green">092</FONT>         * @return the function value.<a name="line.92"></a>
<FONT color="green">093</FONT>         */<a name="line.93"></a>
<FONT color="green">094</FONT>        public double value(double z) {<a name="line.94"></a>
<FONT color="green">095</FONT>           return evaluate(a, c, z);<a name="line.95"></a>
<FONT color="green">096</FONT>        }<a name="line.96"></a>
<FONT color="green">097</FONT>    <a name="line.97"></a>
<FONT color="green">098</FONT>        /**<a name="line.98"></a>
<FONT color="green">099</FONT>         * {@inheritDoc}<a name="line.99"></a>
<FONT color="green">100</FONT>         * @since 3.1<a name="line.100"></a>
<FONT color="green">101</FONT>         */<a name="line.101"></a>
<FONT color="green">102</FONT>        public DerivativeStructure value(final DerivativeStructure t) {<a name="line.102"></a>
<FONT color="green">103</FONT>            verifyInputArray(a, c);<a name="line.103"></a>
<FONT color="green">104</FONT>    <a name="line.104"></a>
<FONT color="green">105</FONT>            final int n = c.length;<a name="line.105"></a>
<FONT color="green">106</FONT>            DerivativeStructure value = new DerivativeStructure(t.getFreeParameters(), t.getOrder(), a[n]);<a name="line.106"></a>
<FONT color="green">107</FONT>            for (int i = n - 1; i &gt;= 0; i--) {<a name="line.107"></a>
<FONT color="green">108</FONT>                value = t.subtract(c[i]).multiply(value).add(a[i]);<a name="line.108"></a>
<FONT color="green">109</FONT>            }<a name="line.109"></a>
<FONT color="green">110</FONT>    <a name="line.110"></a>
<FONT color="green">111</FONT>            return value;<a name="line.111"></a>
<FONT color="green">112</FONT>    <a name="line.112"></a>
<FONT color="green">113</FONT>        }<a name="line.113"></a>
<FONT color="green">114</FONT>    <a name="line.114"></a>
<FONT color="green">115</FONT>        /**<a name="line.115"></a>
<FONT color="green">116</FONT>         * Returns the degree of the polynomial.<a name="line.116"></a>
<FONT color="green">117</FONT>         *<a name="line.117"></a>
<FONT color="green">118</FONT>         * @return the degree of the polynomial<a name="line.118"></a>
<FONT color="green">119</FONT>         */<a name="line.119"></a>
<FONT color="green">120</FONT>        public int degree() {<a name="line.120"></a>
<FONT color="green">121</FONT>            return c.length;<a name="line.121"></a>
<FONT color="green">122</FONT>        }<a name="line.122"></a>
<FONT color="green">123</FONT>    <a name="line.123"></a>
<FONT color="green">124</FONT>        /**<a name="line.124"></a>
<FONT color="green">125</FONT>         * Returns a copy of coefficients in Newton form formula.<a name="line.125"></a>
<FONT color="green">126</FONT>         * &lt;p&gt;<a name="line.126"></a>
<FONT color="green">127</FONT>         * Changes made to the returned copy will not affect the polynomial.&lt;/p&gt;<a name="line.127"></a>
<FONT color="green">128</FONT>         *<a name="line.128"></a>
<FONT color="green">129</FONT>         * @return a fresh copy of coefficients in Newton form formula<a name="line.129"></a>
<FONT color="green">130</FONT>         */<a name="line.130"></a>
<FONT color="green">131</FONT>        public double[] getNewtonCoefficients() {<a name="line.131"></a>
<FONT color="green">132</FONT>            double[] out = new double[a.length];<a name="line.132"></a>
<FONT color="green">133</FONT>            System.arraycopy(a, 0, out, 0, a.length);<a name="line.133"></a>
<FONT color="green">134</FONT>            return out;<a name="line.134"></a>
<FONT color="green">135</FONT>        }<a name="line.135"></a>
<FONT color="green">136</FONT>    <a name="line.136"></a>
<FONT color="green">137</FONT>        /**<a name="line.137"></a>
<FONT color="green">138</FONT>         * Returns a copy of the centers array.<a name="line.138"></a>
<FONT color="green">139</FONT>         * &lt;p&gt;<a name="line.139"></a>
<FONT color="green">140</FONT>         * Changes made to the returned copy will not affect the polynomial.&lt;/p&gt;<a name="line.140"></a>
<FONT color="green">141</FONT>         *<a name="line.141"></a>
<FONT color="green">142</FONT>         * @return a fresh copy of the centers array.<a name="line.142"></a>
<FONT color="green">143</FONT>         */<a name="line.143"></a>
<FONT color="green">144</FONT>        public double[] getCenters() {<a name="line.144"></a>
<FONT color="green">145</FONT>            double[] out = new double[c.length];<a name="line.145"></a>
<FONT color="green">146</FONT>            System.arraycopy(c, 0, out, 0, c.length);<a name="line.146"></a>
<FONT color="green">147</FONT>            return out;<a name="line.147"></a>
<FONT color="green">148</FONT>        }<a name="line.148"></a>
<FONT color="green">149</FONT>    <a name="line.149"></a>
<FONT color="green">150</FONT>        /**<a name="line.150"></a>
<FONT color="green">151</FONT>         * Returns a copy of the coefficients array.<a name="line.151"></a>
<FONT color="green">152</FONT>         * &lt;p&gt;<a name="line.152"></a>
<FONT color="green">153</FONT>         * Changes made to the returned copy will not affect the polynomial.&lt;/p&gt;<a name="line.153"></a>
<FONT color="green">154</FONT>         *<a name="line.154"></a>
<FONT color="green">155</FONT>         * @return a fresh copy of the coefficients array.<a name="line.155"></a>
<FONT color="green">156</FONT>         */<a name="line.156"></a>
<FONT color="green">157</FONT>        public double[] getCoefficients() {<a name="line.157"></a>
<FONT color="green">158</FONT>            if (!coefficientsComputed) {<a name="line.158"></a>
<FONT color="green">159</FONT>                computeCoefficients();<a name="line.159"></a>
<FONT color="green">160</FONT>            }<a name="line.160"></a>
<FONT color="green">161</FONT>            double[] out = new double[coefficients.length];<a name="line.161"></a>
<FONT color="green">162</FONT>            System.arraycopy(coefficients, 0, out, 0, coefficients.length);<a name="line.162"></a>
<FONT color="green">163</FONT>            return out;<a name="line.163"></a>
<FONT color="green">164</FONT>        }<a name="line.164"></a>
<FONT color="green">165</FONT>    <a name="line.165"></a>
<FONT color="green">166</FONT>        /**<a name="line.166"></a>
<FONT color="green">167</FONT>         * Evaluate the Newton polynomial using nested multiplication. It is<a name="line.167"></a>
<FONT color="green">168</FONT>         * also called &lt;a href="http://mathworld.wolfram.com/HornersRule.html"&gt;<a name="line.168"></a>
<FONT color="green">169</FONT>         * Horner's Rule&lt;/a&gt; and takes O(N) time.<a name="line.169"></a>
<FONT color="green">170</FONT>         *<a name="line.170"></a>
<FONT color="green">171</FONT>         * @param a Coefficients in Newton form formula.<a name="line.171"></a>
<FONT color="green">172</FONT>         * @param c Centers.<a name="line.172"></a>
<FONT color="green">173</FONT>         * @param z Point at which the function value is to be computed.<a name="line.173"></a>
<FONT color="green">174</FONT>         * @return the function value.<a name="line.174"></a>
<FONT color="green">175</FONT>         * @throws org.apache.commons.math3.exception.NullArgumentException if<a name="line.175"></a>
<FONT color="green">176</FONT>         * any argument is {@code null}.<a name="line.176"></a>
<FONT color="green">177</FONT>         * @throws NoDataException if any array has zero length.<a name="line.177"></a>
<FONT color="green">178</FONT>         * @throws DimensionMismatchException if the size difference between<a name="line.178"></a>
<FONT color="green">179</FONT>         * {@code a} and {@code c} is not equal to 1.<a name="line.179"></a>
<FONT color="green">180</FONT>         */<a name="line.180"></a>
<FONT color="green">181</FONT>        public static double evaluate(double a[], double c[], double z) {<a name="line.181"></a>
<FONT color="green">182</FONT>            verifyInputArray(a, c);<a name="line.182"></a>
<FONT color="green">183</FONT>    <a name="line.183"></a>
<FONT color="green">184</FONT>            final int n = c.length;<a name="line.184"></a>
<FONT color="green">185</FONT>            double value = a[n];<a name="line.185"></a>
<FONT color="green">186</FONT>            for (int i = n - 1; i &gt;= 0; i--) {<a name="line.186"></a>
<FONT color="green">187</FONT>                value = a[i] + (z - c[i]) * value;<a name="line.187"></a>
<FONT color="green">188</FONT>            }<a name="line.188"></a>
<FONT color="green">189</FONT>    <a name="line.189"></a>
<FONT color="green">190</FONT>            return value;<a name="line.190"></a>
<FONT color="green">191</FONT>        }<a name="line.191"></a>
<FONT color="green">192</FONT>    <a name="line.192"></a>
<FONT color="green">193</FONT>        /**<a name="line.193"></a>
<FONT color="green">194</FONT>         * Calculate the normal polynomial coefficients given the Newton form.<a name="line.194"></a>
<FONT color="green">195</FONT>         * It also uses nested multiplication but takes O(N^2) time.<a name="line.195"></a>
<FONT color="green">196</FONT>         */<a name="line.196"></a>
<FONT color="green">197</FONT>        protected void computeCoefficients() {<a name="line.197"></a>
<FONT color="green">198</FONT>            final int n = degree();<a name="line.198"></a>
<FONT color="green">199</FONT>    <a name="line.199"></a>
<FONT color="green">200</FONT>            coefficients = new double[n+1];<a name="line.200"></a>
<FONT color="green">201</FONT>            for (int i = 0; i &lt;= n; i++) {<a name="line.201"></a>
<FONT color="green">202</FONT>                coefficients[i] = 0.0;<a name="line.202"></a>
<FONT color="green">203</FONT>            }<a name="line.203"></a>
<FONT color="green">204</FONT>    <a name="line.204"></a>
<FONT color="green">205</FONT>            coefficients[0] = a[n];<a name="line.205"></a>
<FONT color="green">206</FONT>            for (int i = n-1; i &gt;= 0; i--) {<a name="line.206"></a>
<FONT color="green">207</FONT>                for (int j = n-i; j &gt; 0; j--) {<a name="line.207"></a>
<FONT color="green">208</FONT>                    coefficients[j] = coefficients[j-1] - c[i] * coefficients[j];<a name="line.208"></a>
<FONT color="green">209</FONT>                }<a name="line.209"></a>
<FONT color="green">210</FONT>                coefficients[0] = a[i] - c[i] * coefficients[0];<a name="line.210"></a>
<FONT color="green">211</FONT>            }<a name="line.211"></a>
<FONT color="green">212</FONT>    <a name="line.212"></a>
<FONT color="green">213</FONT>            coefficientsComputed = true;<a name="line.213"></a>
<FONT color="green">214</FONT>        }<a name="line.214"></a>
<FONT color="green">215</FONT>    <a name="line.215"></a>
<FONT color="green">216</FONT>        /**<a name="line.216"></a>
<FONT color="green">217</FONT>         * Verifies that the input arrays are valid.<a name="line.217"></a>
<FONT color="green">218</FONT>         * &lt;p&gt;<a name="line.218"></a>
<FONT color="green">219</FONT>         * The centers must be distinct for interpolation purposes, but not<a name="line.219"></a>
<FONT color="green">220</FONT>         * for general use. Thus it is not verified here.&lt;/p&gt;<a name="line.220"></a>
<FONT color="green">221</FONT>         *<a name="line.221"></a>
<FONT color="green">222</FONT>         * @param a the coefficients in Newton form formula<a name="line.222"></a>
<FONT color="green">223</FONT>         * @param c the centers<a name="line.223"></a>
<FONT color="green">224</FONT>         * @throws org.apache.commons.math3.exception.NullArgumentException if<a name="line.224"></a>
<FONT color="green">225</FONT>         * any argument is {@code null}.<a name="line.225"></a>
<FONT color="green">226</FONT>         * @throws NoDataException if any array has zero length.<a name="line.226"></a>
<FONT color="green">227</FONT>         * @throws DimensionMismatchException if the size difference between<a name="line.227"></a>
<FONT color="green">228</FONT>         * {@code a} and {@code c} is not equal to 1.<a name="line.228"></a>
<FONT color="green">229</FONT>         * @see org.apache.commons.math3.analysis.interpolation.DividedDifferenceInterpolator#computeDividedDifference(double[],<a name="line.229"></a>
<FONT color="green">230</FONT>         * double[])<a name="line.230"></a>
<FONT color="green">231</FONT>         */<a name="line.231"></a>
<FONT color="green">232</FONT>        protected static void verifyInputArray(double a[], double c[]) {<a name="line.232"></a>
<FONT color="green">233</FONT>            if (a.length == 0 ||<a name="line.233"></a>
<FONT color="green">234</FONT>                c.length == 0) {<a name="line.234"></a>
<FONT color="green">235</FONT>                throw new NoDataException(LocalizedFormats.EMPTY_POLYNOMIALS_COEFFICIENTS_ARRAY);<a name="line.235"></a>
<FONT color="green">236</FONT>            }<a name="line.236"></a>
<FONT color="green">237</FONT>            if (a.length != c.length + 1) {<a name="line.237"></a>
<FONT color="green">238</FONT>                throw new DimensionMismatchException(LocalizedFormats.ARRAY_SIZES_SHOULD_HAVE_DIFFERENCE_1,<a name="line.238"></a>
<FONT color="green">239</FONT>                                                     a.length, c.length);<a name="line.239"></a>
<FONT color="green">240</FONT>            }<a name="line.240"></a>
<FONT color="green">241</FONT>        }<a name="line.241"></a>
<FONT color="green">242</FONT>    <a name="line.242"></a>
<FONT color="green">243</FONT>    }<a name="line.243"></a>




























































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